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  • A semigroup is finite if and only if it is chain-finite and antichain-finite [Elektronski vir]
    Banakh, Iryna ; Banakh, Taras, 1968- ; Bardyla, Serhii
    A subset ▫$A$▫ of a semigroup ▫$S$▫ is called a ▫$chain$▫ (▫$antichain$▫) if ▫$xy\in\{x,y\}$▫ (▫$xy\notin\{x,y\}$▫) for any (distinct) elements ▫$x,y\in S$▫. A semigroup ▫$S$▫ is called ... (▫$anti$▫)▫$chain$▫-▫$finite$▫ if ▫$S$▫ contains no infinite (anti)chains. We prove that each antichain-finite semigroup ▫$S$▫ is periodic and for every idempotent ▫$e$▫ of ▫$S$▫ the set ▫$\sqrt[\infty]{e}=\{x\in S:\exists n\in\mathbb N\;\;(x^n=e)\}$▫ is finite. This property of antichain-finite semigroups is used to prove that a semigroup is finite if and only if it is chain-finite and antichain-finite. Also we present an example of an antichain-finite semilattice that is not a union of finitely many chains.
    Vir: Axioms [Elektronski vir]. - ISSN 2075-1680 (Vol. 10, iss. 1, March 2021, art. 9 (6 str.))
    Vrsta gradiva - e-članek
    Leto - 2021
    Jezik - angleški
    COBISS.SI-ID - 47500291

    Povezava(-e):

    https://doi.org/10.3390/axioms10010009
    DOI

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